## Optimized Windows

We close this chapter with a general discussion of *optimal
windows* in a wider sense. We generally desire

(4.59) |

but the nature of this approximation is typically determined by characteristics of audio perception. Best results are usually obtained by formulating this as an

*FIR filter design problem*(see Chapter 4). In general, both time-domain and frequency-domain specifications are needed. (Recall the potentially problematic impulses in the Dolph-Chebyshev window shown in Fig.3.33 when its length was long and ripple level was high). Equivalently, both

*magnitude*and

*phase*specifications are necessary in the frequency domain.

A window transform can generally be regarded as the frequency response
of a *lowpass filter* having a *stop band* corresponding to
the side lobes and a *pass band* corresponding to the main lobe
(or central section of the main lobe). Optimal lowpass filters
require a *transition region* from the pass band to the stop
band. For spectrum analysis windows, it is natural to define the
*entire main lobe* as ``transition region.'' That is, the
pass-band width is zero. Alternatively, the pass-band could be
allowed to have a finite width, allowing some amount of ``ripple'' in
the pass band; in this case, the pass-band ripple will normally be
maximum at the main-lobe midpoint (
, say), and at the
pass-band edges (
). By
embedding the window design problem within the more general problem of
FIR digital filter design, a plethora of optimal design techniques can
be brought to bear
[204,258,14,176,218].

### Optimal Windows for Audio Coding

Recently, numerically optimized windows have been developed by Dolby which achieve the following objectives:

- Narrow the window in time
- Smooth the onset and decay in time
- Reduce side lobes below the
*worst-case masking threshold*

See §4.10 for an overview of optimal methods for FIR digital filter design.

### General Rule

There is rarely a closed form expression for the optimal window in
practice. The most important task is to formulate an *ideal
error criterion*. Given the right error criterion, it is usually
straightforward to minimize it numerically with respect to the window
samples
.

**Next Section:**

Window Design by Linear Programming

**Previous Section:**

Gaussian Window and Transform